CMG Collaborative Research: Fast and Efficient Radial Basis Function Algorithms for Geophysical Modeling on Arbitrary Geometries

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Numerical method innovations in the geosciences have not paralleled theexplosion in computer hardware development over the last decades. Yet, forscientific computing to advance, it is crucial that novel numericalapproaches are developed that improve the simplicity, flexibility, andaccuracy of algorithms while taking advantage of this revolution in hardwaretechnology. The current project is aimed at exactly that objective: todevelop fast, efficient and parallelizable radial basis function (RBF)methodologies, enhanced by novel graphics processing unit (GPU) technology,for applications in computational geosciences. A collaborative team ofcomputational mathematicians and geophysicists has been established for thispurpose. The RBF approach is very attractive in that it achieves high-orderaccuracy for arbitrary geometries in n-dimensional space, naturally permitslocal refinement, is mesh-free (no grids), algorithmic complexity does notincrease with dimension and, generally offers higher accuracy with longertime steps than traditional spectral methods for a given number of nodes.However, RBFs are still in an early developmental stage; key issues to beaddressed are: 1) development of localized high-order RBF-finite difference(RBF-FD) stencils on irregular node layouts in n-dimensional space forarbitrary geometries with stable time-stepping, 2) adaptive local noderefinement schemes for RBF-FD, 3) development of hybrid spectral RBF and FDschemes to maximize advantages of RBFs and while minimizing theircomputational cost, and 4) implementation of the method on hardwareaccelerators, such as GPUs. Scientific targets for application include 3Dmantle convection with varying properties, models of the geodynamo in anelliptic geometry, and tsunami modeling with irregular coastline topography.By advancing the frontier of computational mathematics that will takeadvantage of today?s booming technology industry, society can be offered amore in depth understanding of geophysical processes related to mantleconvection, the geodynamo, and tsunamis. These phenomena play a key role incontinental movement, polarity reversal of the earth?s magnetic field,earthquakes, and coastal flooding. The proposed innovative mathematicalcomputational approach may hold the key to unlocking long-standing questions of fundamental importance in such geophysics areas. An interdisciplinary collaborative team of computational mathematicians and geophysicists has been assembled from a national lab and four universities (NCAR-Colorado, Boise State Univ.-Idaho, Univ. of Minnesota, Florida State Univ., and the Univ. of California?Davis) to develop new methods in computational mathematics for addressing these critical geophysical problems. The project supports collaborative participation among researchers at different stages in their careers as well as Ph.D, Masters, and undergraduate students in 4 states and in a myriad of scientific and mathematical disciplines. These students will have the opportunity to participate as members of an interdisciplinary team composed of senior personnel with a demonstrated commitment to education and research. NCAR will serve not only as an integrating hub for scientific endeavors but provide a medium for students from across the country to work together on multiple facets of the proposal. Through the SciPARCS 10 week internship program at NCAR, students will have the opportunity to engage in joint collaborative research, preparing them for a career in computational geosciences for the 21st century.